Black-body radiation: Difference between revisions

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{{unreferenced|date=March 2013}}
{{for|the topological usage|Rose (topology)}}
[[File:7_Petal_rose.svg|thumb|200px|right|Rose with ''k'' = 7 petals]]
[[File:8-Petal_rose.svg|thumb|200px|right|Rose with 8 petals (''k''=4).]]
[[File:Rose-rhodonea-curve-7x9-chart-improved.svg|thumb|200px|right|Rose curves defined by
<math>r=\cos k \theta</math>, for various values of ''k''=n/d.]]
<!-- [[Image:7-leafed rose.gif|thumb|200px|right|Rose with ''k'' = 7 petals]] -->
<!-- [[Image:8-leafed rose.gif|thumb|200px|right|Rose with 8 petals (''k''=4).]] -->


In [[mathematics]], a '''rose''' or '''rhodonea curve''' is a [[sine wave|sinusoid]] plotted in [[polar coordinates]].


== General overview ==
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Up to [[Similarity (geometry)|similar]]ity, these curves can all be expressed by a polar equation of the form
:<math>\!\,r=\cos(k\theta)</math>
or, alternatively, as a pair of Cartesian parametric equations of the form
:<math>\!\,x=\cos(kt)\sin(t)</math>
:<math>\!\,y=\cos(kt)\cos(t)</math>
If ''k'' is an integer, the curve will be rose shaped with
*2''k'' petals if ''k'' is even, and
*''k'' petals if ''k'' is odd.
When ''k'' is even, the entire graph of the rose will be traced out exactly once when the value of θ changes from 0 to 2π.   When ''k'' is odd, this will happen on the interval between 0 and π. (More generally, this will happen on any interval of length 2π for ''k'' even, and π for ''k'' odd.)
 
If ''k'' ends in 1/2 (ex: 0.5, 2.5), the curve will be rose shaped with 4''k'' petals.
 
If ''k'' ends in 1/6 or 5/6 and is greater than 1 (ex: 1.16666667, 2.8333333), the curve will be rose shaped with 12''k'' petals.
 
If ''k'' ends in 1/3 and is greater than 1 (ex: 1.333333, 2.333333), the curve will be rose shaped and will have:
*3''k'' petals if ''k'' is even, and
*6''k'' petals if ''k'' is odd.
 
If ''k'' ends in 2/3 and is greater than 1 (ex: 1.666667, 2.666667), the curve will be rose shaped and will have:
*6''k'' petals if ''k'' is even, and
*3''k'' petals if ''k'' is odd.
 
If ''k'' is [[Rational number|rational]], then the curve is [[Closed (mathematics)|closed]] and has finite length.  If ''k'' is [[Irrational number|irrational]], then it is not closed and has infinite length.  Furthermore, the graph of the rose in this case forms a [[dense set]] (i.e., it comes arbitrarily close to every point in the unit disk).
 
Since
:<math>\sin(k \theta) = \cos\left( k \theta - \frac{\pi}{2} \right) = \cos\left( k \left( \theta-\frac{\pi}{2k} \right) \right)</math>  
for all <math>\theta</math>, the curves given by the polar equations
:<math>\,r=\sin(k\theta)</math> and <math>\,r = \cos(k\theta)</math>
are identical except for a rotation of π/2''k'' radians.
 
Rhodonea curves were named by the Italian mathematician [[Guido Grandi]] between the year 1723 and 1728.<ref>{{MacTutor Biography|class=Curves|id=Rhodonea|title=Rhodonea}}</ref>
 
==Area==
A rose whose polar equation is of the form
:<math>r=a \cos (k\theta)\,</math>
where ''k'' is a positive integer, has [[Polar coordinates#Integral calculus|area]]
:<math>
    \frac{1}{2}\int_{0}^{2\pi}(a\cos (k\theta))^2\,d\theta = \frac {a^2}{2} \left(\pi + \frac{\sin(4k\pi)}{4k}\right) = \frac{\pi a^2}{2}
</math>
if ''k'' is even, and
:<math>
    \frac{1}{2}\int_{0}^{\pi}(a\cos (k\theta))^2\,d\theta = \frac {a^2}{2} \left(\frac{\pi}{2} + \frac{\sin(2k\pi)}{4k}\right) = \frac{\pi a^2}{4}
</math>
if ''k'' is odd.
 
The same applies to roses with polar equations of the form
:<math>r=a \sin (k\theta)\,</math>
since the graphs of these are just rigid rotations of the roses defined using the cosine.
 
==How the parameter k affects shapes==
In the form ''k''&nbsp;=&nbsp;''n'', for integer n, the shape will appear similar to a flower. If ''n'' is odd half of these will overlap, forming a flower with ''n'' petals. However if it is even the petals will not overlap, forming a flower with 2''n'' petals.
 
When ''d'' is a prime number then ''n''/''d'' is a least common form and the petals will stretch around to overlap other petals. The number of petals each one overlaps is equal to the how far through the sequence of primes this prime is +1, i.e. 2 is 2, 3 is 3, 5 is 4, 7 is 5, etc.
 
In the form ''k''&nbsp;=&nbsp;1/''d'' when ''d'' is even then it will appear as a series of ''d''/2 loops that meet at 2 small loops at the center touching (0,&nbsp;0) from the vertical and is symmetrical about the ''x''-axis.
If ''d'' is odd then it will have ''d''&nbsp;div&nbsp;2 loops that meet at a small loop at the center from ether the left (when in the form ''d''&nbsp;=&nbsp;4''n''&nbsp;&minus;&nbsp;1) or the right (''d''&nbsp;=&nbsp;4''n''&nbsp;+&nbsp;1).
 
If ''d'' is not prime and n is not 1, then it will appear as a series of interlocking loops.
 
If ''k'' is an irrational number (e.g. <math>\pi</math>, <math>\sqrt{2}</math>, etc.) then the curve will have infinitely many petals, and it will be [[Dense set|dense]] in the unit disc.
 
==See also==
*[[Lissajous curve]]
*[[quadrifolium]] - a rose curve with ''k'' = 2.
*[[Maurer rose]]
*[[Rose (topology)]]
 
==Notes==
<references />
 
==External links==
{{commonscat|Rose curves}}
* {{MathWorld|Rose|Rose}}
* [http://www25.brinkster.com/denshade/PolarFlower.html Applet to create rose with k parameter]
* [http://xahlee.org/SpecialPlaneCurves_dir/Rose_dir/rose.html Visual Dictionary of Special Plane Curves] Xah Lee
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rose Interactive example with JSXGraph]
 
[[Category:Curves]]

Revision as of 12:08, 20 February 2014


There are legion great tools these days that is employed to create gardener's life a little easier. Actually most of these experts make tough part is holding job of gardening a great deal easier, faster or more efficient, thus saving the homeowner far of determination.

Before we are able to consider the Qi to flow through your home, need to attract it onto your dwelling and then move it towards your front exit. The relationship between your homes' landscaping and its physical structure is an excellent aspect of Feng Shui. Several different traditional techniques can be used to direct the Qi and then align it with the architectural regarding the creating / developing.



I was nervous in order to Haulover Swimming pool. I didn't know what to be expecting. My wife had been there on several previous occasions with friends. She liked the nude beach for the straightforward reason that they abhorred tan lines, so, as was probably the most relaxing and enjoyable method away all of them. Personally, I didn't care a lick about tan lines, but Experienced been curious on this social taboo.

Another good idea for paddock maintenance is keeping anyone of you paddock neat and tidy. This typically involve keeping the grass reasonably short, and like a general rule you should aim for approx 5cm in summer and 2cm in winter months. This may well to prevent weeds from going to seed. hedges should normally be trimmed in the autumn and also places is essential to keep all fences in great.

Water features are a good way to help increase and redirect the flow of energy to best door. If possible, put a water fountain or feature somewhere near you gate. Try to set up the fountain therefore the water isn't pouring vertically. Instead, choose a fountain or handcrafted feature whereas the water is flowing towards the front door of your house before the pump actually starts to re-circulate normal water. The direction of the waterflow and drainage is important, because the Qi become attracted towards the movement of the water additionally helps make certain that a controlled flow of one's to the top door of your property.

There is convincing evidence that buyers make the minds of men up the particular first 1/2 minute or so, when first viewing a property. That's why the front walkway, steps, front door, and entrance are so important. Even if your master bedroom on develop floor is gorgeous, in the event a buyer is turned off by period they get upstairs, actual too past. Make sure your front walkway is see-through of weeds and debris (and ice in the winter!), do your best to make the doorway clean and tidy (put away extra shoes and boots), consider painting best door (nice and !) - creates a great first impression; consider replacing a rusted mailbox, . . .. If your front screen door is torn, replace the screen.

In this process, some branches which are situated nearby the crown for this tree are removed. This is done for the trees at the footpaths and roads. May help the better light penetration and raises the scenery just about.

If you have often seen Star Wars: The Clone Wars you'll remember the laser cannons and missiles and the clone turbo tank rolling through separatist opposition. And also the way Cad Bane has other plans as he sabotages them from right under the Jedi Knights. This set helps you relive that and the characters Anakin, Ahsoka, and Aayla Secura are normally part of their 1139 pieces and 5 minifigures!

When you have just about any questions relating to where along with how to make use of hedgingplants.com, you can contact us on our own website.

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